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Binomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Binomial[n,k] > Summation > Infinite summation





http://functions.wolfram.com/06.03.23.0036.01









  


  










Input Form





Sum[k^6/(2^k Binomial[2 k, k]), {k, 1, Infinity}] == 1288944/117649 + (2822168 ArcCot[Sqrt[7]])/(117649 Sqrt[7])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["k", "6"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", "k"]], "]"]]]]], "\[Equal]", RowBox[List[FractionBox["1288944", "117649"], "+", FractionBox[RowBox[List["2822168", " ", RowBox[List["ArcCot", "[", SqrtBox["7"], "]"]]]], RowBox[List["117649", " ", SqrtBox["7"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> k </mi> <mn> 6 </mn> </msup> </mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;k&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1288944 </mn> <mn> 117649 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 2822168 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mn> 7 </mn> </msqrt> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 117649 </mn> <mo> &#8290; </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <apply> <ci> Binomial </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 1288944 <sep /> 117649 </cn> <apply> <times /> <cn type='integer'> 2822168 </cn> <apply> <arccot /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 117649 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k_"]]], " ", SuperscriptBox["k_", "6"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "k_"]], ",", "k_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1288944", "117649"], "+", FractionBox[RowBox[List["2822168", " ", RowBox[List["ArcCot", "[", SqrtBox["7"], "]"]]]], RowBox[List["117649", " ", SqrtBox["7"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2002-12-18